If there be no friction, what JustinRyan said is true... the balls keep on sliding forever with equal acceleration.
But then again, the question specifically says that the balls are rolling, and hence that possibility is eliminated. That's because a frictional force is necessary for causing the rolling motion!
If they are released from the same height, at the bottom of the inclination, they have the same velocities.
If there be friction in the ground as well, both the balls will get the same retardation! So shouldn't the balls cover the same distance?
And I don't understand how you use inertia here! I agree that the heavier ball should be harder to stop than the lighter one, but then again, greater frictional force acts on the heavier one!
Please help me understand this...
Also regarding energy changes, the heavier one has initially more energy, so what?

but if we take inertia of rest for a heavier ball , it slows down and stops.

Nope. Object in motion will stay in motion, barring any forces. Inertia means it will stay in motion, not slow down. I wouldn't try to distinguish between inertia of rest and inertia of motion. Best just to think of inertia.

Friction is only required to set the ball into rotation. Once the ball reaches the end of the incline its tendency will be to retain its rotational and translational motions. The only force opposing this tendency is air resistance as all other forces are normal to the motion. If the balls are of similar size and surface finish then air resistance will be equal for both balls. Since this equates to a larger deceleration for the lighter ball.....
If this is clearly untrue....enlighten me please

Nope. Object in motion will stay in motion, barring any forces. Inertia means it will stay in motion, not slow down. I wouldn't try to distinguish between inertia of rest and inertia of motion. Best just to think of inertia.

He means the inverse, which is also true and is described by Newton's equations--Any object at rest will stay at rest unless acted upon by a different force. So the heavier object will have a harder time starting up than the lighter.

The thing here is that because of the same law he cited, the heavier object will take longer to stop than the lighter.

If all forces are equal and opposite, then the two parts of the law will equate with each other if the balls come back to rest at another place with the same inclination. Thus, the balls will stop at the same place.

The problem is, the OP hasn't given enough specifics in order for us to answer the question accurately.

If we assume that as the ball is going down the slope, the energy lost to dissipation is negligible, then since [itex]I=2/5MR^2[/itex] and [itex]KE=1/2MI \omega^2[/itex], both balls will be at the same speed when they reach the bottom of the slope.

So now we need to decide what kind of resistance to motion acts on the balls once they are travelling horizontally along the flat part. If it is rolling resistance, then the resistance force is proportional to the normal force (and therefore mass) of the ball. So the power lost is proportional to mass, and the initial energy of the ball in the horizontal section is also proportional to the mass. Therefore, both balls stop after the same distance in this case.

We could instead assume that most of the dissipative resistance comes from air resistance. The most basic model of air resistance says that the resistive force is proportional to speed squared and not dependent on the mass of the object. So in this case, the denser ball will go ahead of the less dense ball, since it has greater inertia than the less dense ball.

The most basic model of air resistance says that the resistive force is proportional to speed squared and not dependent on the mass of the object. So in this case, the denser ball will go ahead of the less dense ball, since it has greater inertia than the less dense ball.

This statement of yours is self-contradicting!
First you say that Fair resistance[itex]\propto[/itex]v2 and independant of mass!
Then you say that the denser ball keeps going further... because of inertia,ie, beacause of mass??
Previously you said (and I agree) that both the balls have same speed when they reach the bottom.
So then Fair resistance should be equal for both of them.

One more thing, if we are considering air resistance, then we should also consider it while the balls are rolling down the incline, which nobody stated before! The result will be an unnecessary complexity to this simple "innocent" question.

Why go into such troubles? Fair resistance will be negligible if the balls are small and moving with relatively small speeds!

I believe that the only retardation that will make a difference is the frictional force from the ground. But since this Fground friction is dependant on weight an the co-eff. of ground friction, its simply Fground friction=μmg.
So the retardation is simply: [itex]\frac{F}{m}[/itex]=μg.
So it all comes down to whether or not they have same μ. If they do, they will definitely stop after the same distance. Else, the one with lesser μ will go further!
Don't hesitate to discard all this if I'm worng.

There is nothing contradictory there. The air resistance is not dependent on mass so each ball given they are the same size and have the same velocity will experience the same resistive force. If this is equal for both balls the effect will be greater for the lighter ball as this is dependant on mass.
I think that as you say the air resistance can be minimised as can the rolling resistance by having a smooth track and ball surface. The results of the experiment will depend on these conditions and which type of resistance is more significant. As has been stated already there has not been enough information given to determine the outcome.

This statement of yours is self-contradicting!
First you say that Fair resistance[itex]\propto[/itex]v2 and independant of mass!
Then you say that the denser ball keeps going further... because of inertia,ie, beacause of mass??

This is no contradiction. If both balls are moving at some speed, then the force due to air resistance is the same on both, therefore since the denser ball has greater inertia, air resistance will have less of an effect on it. And yes, the inertia will be the inertia of a sphere, which is proportional to the mass of the ball.

deep383 said:

Previously you said (and I agree) that both the balls have same speed when they reach the bottom.
So then Fair resistance should be equal for both of them.

Yeah, I said that both balls have same speed when they reach the bottom of the incline. So then when they travel along the horizontal part, the deceleration of the denser ball will be less, so the denser ball goes further on the horizontal part. (This is if we assume air resistance is most important).

deep838 said:

One more thing, if we are considering air resistance, then we should also consider it while the balls are rolling down the incline, which nobody stated before! The result will be an unnecessary complexity to this simple "innocent" question.

I made the assumption that the incline is short enough compared to the horizontal section, so that the speeds of the two balls when they get to the bottom of the incline is approximately the same. We could instead say that air resistance affects then on the incline as well. In that case, the denser ball would simply get ahead further. And so we get the same qualitative answer.

deep838 said:

I believe that the only retardation that will make a difference is the frictional force from the ground. But since this Fground friction is dependant on weight an the co-eff. of ground friction, its simply Fground friction=μmg.
So the retardation is simply: [itex]\frac{F}{m}[/itex]=μg.
So it all comes down to whether or not they have same μ. If they do, they will definitely stop after the same distance. Else, the one with lesser μ will go further!
Don't hesitate to discard all this if I'm worng.

Yeah, well that is one possible case. But we can also imagine a situation where the air resistance is more important. As others have said, we would need more information to be able to decide if air resistance or rolling resistance is more important.

Also, I should say that the retardation due to contact with the ground is called rolling resistance (not usually called frictional force from the ground). This is because the frictional force from the ground does not necessarily cause dissipation of energy. It is only when the ball is slipping with respect to the ground, that friction causes dissipation. There are also other effects which cause dissipation due to contact with the ground, which is why the term 'rolling resistance' is used. (wikipedia have a useful page about it).

JustinRyan:"..If this is equal for both balls the effect will be greater for the lighter ball as this is dependant on mass..."
What do you mean? What effect? Isn't the effect the air resistance? If it doesn't depend on mass, then the effect will be equal. But it does, the drag is dependent on the speed and density of object, and therefore on mass.

BruceW i wrote the 1st part wrong, I only realised that after I saw the replies, sorry for that.
And I wasn't familiar with rolling resistance! I thought that friction always acts as a dissipative force, but you say that i behaves so only sometimes. I give no comments on that for now since I know nothing about it.
When I saw this thread, I thought it will have a simple solution, but I'm forced to think otherwise now! This problem seemed so easy and basic "which will cover more distance?" something every kid asks! But I have to say that I'm feeling really disappointed about it. :(

yeah, it is quite complicated because the problem involves dissipation, which often makes physics problems more complicated. The easy bit is when we assume zero dissipation. Then when we want to change our model to include dissipation, it usually gets more complicated.

Also, about friction, we can think about two surfaces in contact with each other. If the two surfaces have zero motion relative to each other, where they are in contact, then there is zero dissipation of energy. It is only when the two surfaces slide past each other that energy is dissipated.

A wheel is a very good example. If the wheel is rolling without slipping, then the part of the wheel touching the ground has zero motion relative to the ground. (Which might be counter-intuitive when you first hear it, but draw a diagram and think about it for a bit, and you'll realise it is true). Therefore, there is zero energy lost by friction when the wheel is rolling without slipping. So in this case, the rolling resistance must be due to other causes, not simply friction with the ground.